Simple Geometric Control Algorithm and Computation of Lever Arms

The prior considerations directly yield the basis for a simple, roll angle based control algorithm incorporated in the PMC. While the target lateral offset Δytarget of BSTAM was defined in eq. (4.5), its limitations are defined as:

(4.8) e₂

Dy_BSTAM

4.2 Mechanical Setup of the BSTAM Prototype

(4.9)

From that, the BSTAM actuation angle can be computed as:

ε (4.10)

and the according longitudinal offset of the BSTAM is then given by:

(4.11)

This variation in upper fork yoke offset is affecting caster angle and trail as follows:

(4.12)

with bd being the bearing distance along z’st-coordinates, yielding the effective caster

angle, as illustrated in Figure 4.8, right:

(4.13)

Again neglecting small changes Δh1 in the vertical distance h1, the frontally projected

king-pin and steering axis inclination angles σ and γ as introduced in Figure 3.1 are:

(4.14)

(4.15)

The compensated portion of the tire scrub radius srcmp and the resulting effective com- pensation ratio ecr according to the same figure are then given by:

(4.16)

(4.17)

using srtir from eq. (4.4) and ΔyBSTAM from eq. (4.9).

The effective scrub radius towards the inclined BSTAM steering axis is then:

(4.18)

while the effective trail neff can be computed within triangle (N-G-O) in Figure 4.8:

(4.19)

yielding the effective normal trail nteff:

Finally, the effective lever arms of front tire forces towards the inclined steering axis (as already used for simulations in chapter 3) are computed in analogy to eq. (3.8) to (3.10): (4.21)

_(4.22)

_(4.23)

Figure 4.11: BSTAM excenter angle and effective compensation ratio as a function of roll angle, target compensation ratio, fork travel and pitch. Black lines for correct chassis parame- ters, grey lines as erroneously incorporated into the control algorithm of the prototype.

The BSTAM actuation angle obtained from the presented control algorithm as well as the achievable effective compensation ratio are illustrated as a function of roll angle,

4.2 Mechanical Setup of the BSTAM Prototype

While the black lines indicate the use of correct chassis parameters (bd = 233 mm, fo = 30 mm, rft = 295 mm, and cos(τ) ≈ 0.914), a preliminary version of the control algorithm was kept in the prototype motorcycle due to a programming error, yielding the grey curves. It uses different projection parameters (bd = 233.5 mm, fo = 0, rft = 282 mm, and cos(τ) ≈ 0.999, the latter as a mismatch between computation in radi- an and degree), resulting in slightly higher actuation angles and compensation ratios. The example shows the limit roll angles, at which a certain compensation target can still be achieved. E.g. for λ = 35° in the last diagram, the maximal achievable ecr is about 0.6, no matter how high the target compensation ratio tcr is chosen.

4.2.4 Chassis Geometry Changes through BSTAM

Finally, a parameter variation of BSTAM actuation angle, roll angle, fork travel and pitch reveals the maximal possible chassis parameter variations through BSTAM as presented in Figure 4.12 as well as Figure 2.21.

Figure 4.12: Chassis parameter variations due to BSTAM actuation, roll, fork travel, and pitch. (Each annotation in the left diagram applies to one horizontal black Standard and two curved grey BSTAM graph lines, marked with an arrow at their respective intersection.)

The reference position of the BSTAM actuation angle is ε = 0, when the kinematic Straight 60° Roll Straight, 10° Pitch 60° Roll, 10° Pitch

angle and trail (cf. Figure 4.7, Figure 4.8, and Figure 2.21), hence called a “long trail” setup. Consequently, ε = 90° is equal to full lateral displacement, putting the steering axis back to its original plane (y’st-z’st) and yielding the same trail values as for the

baseline chassis (cf. Figure 4.12, left). Finally, ε = 180° represents a “short trail” setup, with the kinematic center in front of the steering shaft. While steering transmission and ease of handling are compromised for the long trail setup along with an increasing stability (cf. chapters 2.1.5 and 2.1.6), the opposite is the case for the short trail setup. The left illustration in Figure 4.12 shows, how the trail in straight running increases from 98 mm of the reference towards 128 mm for the long trail setup and decreases to only 68 mm for the short trail setup. While fork compression decreases the trail value in long trail setups, it is increasing them for the short trail ones, which is favorable in both cases. Following the diagram from top to bottom shows, how the trail is already de- creasing for the standard setup with increasing roll and pitch (of cause with an exagger- ated racing style roll angle of λ = 60° to highlight the effect) to a value as low as 34 mm. Given a BSTAM with its steering axis passively centered in a short trail setup, trail is decreasing to only 7 mm, which is theoretically critical from a stability point of view. However, during orienting tests with short trail setups, this was practically not an issue. Firstly, roll and pitch angels stayed below these example values. Secondly, the pneumat- ic trail of tires is actually increasing the effective trail, yielding higher stability. And thirdly, when BSTAM is used in active mode, the excenter assumes turn angles close to the pre-set 80° limit for large roll angles. It is hence favorably increasing (decreasing) the trail of a short (long) trail setup to be relatively close to the value of the standard steering geometry, especially at the beginning of a corner braking maneuver. Despite the possibility to actually ride with short trail setups, the riding tests presented in chapter 5 were completely focused on the “safer” long trail setup, in face of the impending winter. The right side of Figure 4.12 shows the lateral and longitudinal displacements of the BSTAM’s kinematic center point as well as its lateral steering axis inclination in the fork coordinate system and the variations in caster angle.

While the general steering axis inclination with regards to the steering shaft is equal to the maximal changes in caster angle:

(4.24)

the frontal projection of the same at maximal lateral displacement is:

(4.25)

When limited to 80° actuation angle, these values are only marginally reduced in the order of 1% and become:

4.3 Overview of the BSTAM Prototype (4.26) (4.27) Which underlines the assumption of negligible changes in the transfer ratio of second- ary effects on steering torque demand as discussed in chapter 3.3.5.

4.3 Overview of the BSTAM Prototype

Figure 4.13: Overview of the ready BSTAM Prototype Motorcycle (PMC) Setup with different center of gravity locations

Figure 4.13 gives an overview of the BSTAM Prototype Motorcycle (PMC) ready for testing. The numbers indicate the center of gravity location in different vehicle setups, that can be read from Table 4.4 along with the respective mass increase that was plus